When constructing a scalar field theory we provide an action with Lagrangian describing the structure of the field. In a curved spacetime the kinetic term derivatives are still partial. Does the fact that the integral "sees" infinitesimal neighborhood of a volume element, where spacetime is flat, is responsible for the type of derivatives?
Dear Lukasz, Covariant derivative generally has been defined as an operator action on a "Tensor field" on a manifold. Whenever operator acts on a "scalar" field (zero order tensor field) it results in the regular partial derivative, because scalar fields do not depend on "basis vectors" and hence the additional terms containing connection (i.e. Levi civita ) which appears in covariant derivatives of vectors and tensors, would be absent in scalar case. In other words covariant derivatives of scalar fields are the same as regular partial derivatives, then if we define a scalar field even on a curved space-time, its kinetic terms will be presented as quadratic of partial derivatives of scalars. However if we impose a gauge symmetry, the gauge covariant derivative will appear in Lagrangian density with resultant additional terms of interactions.
Dear Manouchehr,
Thank you very much for the answer. It illuminated the topic brilliantly!
- Badges
- Science topic